Expansion Complexes for Finite Subdivision Rules Ii
نویسنده
چکیده
This paper gives applications of earlier work of the authors on the use of expansion complexes for studying conformality of finite subdivision rules. The first application is that a one-tile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching 0) has an invariant conformal structure, and hence is conformal. The paper next considers one-tile single valence finite subdivision rules. It is shown that an expansion map for such a finite subdivision rule can be conjugated to a linear map, and that the finite subdivision rule is conformal exactly when this linear map is either a dilation or has eigenvalues that are not real. Finally, an example is given of an irreducible finite subdivision rule that has a parabolic expansion complex and a hyperbolic expansion complex.
منابع مشابه
Expansion complexes for finite subdivision rules
This paper develops the basic theory of conformal structures on finite subdivision rules. The work depends heavily on the use of expansion complexes, which are defined and discussed in detail. It is proved that a finite subdivision rule with bounded valence and mesh approaching 0 is conformal (in the combinatorial sense) if there is a partial conformal structure on the model subdivision complex...
متن کاملCombinatorially Regular Polyomino Tilings
Let T be a regular tiling of R which has the origin 0 as a vertex, and suppose that φ : R → R is a homeomorphism such that i) φ(0) = 0, ii) the image under φ of each tile of T is a union of tiles of T , and iii) the images under φ of any two tiles of T are equivalent by an orientation-preserving isometry which takes vertices to vertices. It is proved here that there is a subset Λ of the vertice...
متن کاملConstructing Subdivision Rules from Rational Maps
This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f is a critically finite rational map with no periodic critical points, then for any sufficiently large integer n the iterate f is the subdivision map of a finite subdivision rule. We are interested here in connections between finite subdivision rules and rational...
متن کاملClassification of Subdivision Rules for Geometric Groups of Low Dimension
Subdivision rules create sequences of nested cell structures on CWcomplexes, and they frequently arise from groups. In this paper, we develop several tools for classifying subdivision rules. We give a criterion for a subdivision rule to represent a Gromov hyperbolic space, and show that a subdivision rule for a hyperbolic group determines the Gromov boundary. We give a criterion for a subdivisi...
متن کاملStiefel-whitney Homology Classes of Quasi-regular Cell Complexes
A quasi-regular cell complex is defined and shown to have a reasonable barycentric subdivision. In this setting, Whitney's theorem that the ^-skeleton of the barycentric subdivision of a triangulated n-manifold is dual to the (n /c)th Stiefel-Whitney cohomology class is proven, and applied to projective spaces, lens spaces and surfaces. 1. QR complexes. A (finite) cell structure on a space X is...
متن کامل